A242939 -id:A242939 - cp's OEIS frontend

A182176 Number of affine subspaces of GF(2)^n.

1, 3, 11, 51, 307, 2451, 26387, 387987, 7866259, 221472147, 8703733139, 479243212179, 37070813107603, 4036214347068819, 619402703369958803, 134108807406166799763, 40994263184865380595091, 17700624176280878586721683, 10799420012335823235718509971

Offset: 0

Gaëtan Leurent, Apr 16 2012

q-binomial transform of A000079 for q=2. - Vladimir Reshetnikov, Oct 17 2016
From Geoffrey Critzer, Jul 15 2017: (Start)
a(n) is the total number of vectors in all subspaces of GF(2)^n.
a(n) is the number of subspaces of GF(2)^(n+1) that do not contain a given nonzero vector. (End)

			For n=2, there are 4 affine subspaces of dimension 0, 6 of dimension 1, and 1 of dimension 2.

Gaëtan Leurent, Table of n, a(n) for n = 0..100
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.

Cf. A006116.

List([0..20],n->Sum([0..n],k->(2^n/2^k*Product([0..k-1],i->(2^n-2^i)/(2^k-2^i))))); # Muniru A Asiru, Aug 01 2018

Table[Sum[2^n/2^k * Product[(2^n-2^i)/(2^k-2^i),{i,0,k-1}],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 22 2014 *)
Table[Sum[QBinomial[n, k, 2] 2^k, {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 17 2016 *)

{a(n)=polcoeff(sum(m=0,n,x^m/prod(k=1,m+1,1-2^k*x+x*O(x^n))),n)} /* Paul D. Hanna, May 01 2012 */

def a(n): return sum([(2^n/2^k)*prod([(2^n-2^i)/(2^k-2^i) for i in [0..k-1]]) for k in [0..n]])

a(n) = Sum_{k=0..n} (2^n/2^k * Product_{i=0..k-1} (2^n - 2^i)/(2^k - 2^i)).
G.f.: Sum_{n>=0} x^n / Product_{k=1..n+1} (1-2^k*x). - Paul D. Hanna, May 01 2012
a(n) ~ c * 2^((n+1)^2/4), where c = EllipticTheta[2, 0, 1/2] / QPochhammer[1/2, 1/2] = A242939 = 7.3719494907662273375414118336... if n is even, and c = EllipticTheta[3, 0, 1/2] / QPochhammer[1/2, 1/2] = A242938 = 7.3719688014613165091531912082... if n is odd. - Vaclav Kotesovec, Jun 22 2014
a(n)/[n]q! is the coefficient of x^n in the expansion of (1 + x)*exp_q( x)*exp_q(x) when q->2 and where exp_q(x) is the q exponential function and [n]_q! is the q-factorial of n. - _Geoffrey Critzer, Jul 15 2017
a(n) = (2^n - 1)*A006116(n-1) + A006116(n). - Geoffrey Critzer, Jul 15 2017

A242938 Decimal expansion of c_e, coefficient associated with the asymptotic evaluation c_e*2^(n^2/4) of the number of subspaces of the n-dimensional vector space over the finite field F_2, n being even.

Original entry on oeis.org

7, 3, 7, 1, 9, 6, 8, 8, 0, 1, 4, 6, 1, 3, 1, 6, 5, 0, 9, 1, 5, 3, 1, 9, 1, 2, 0, 8, 2, 6, 8, 0, 9, 1, 5, 8, 8, 8, 5, 8, 7, 6, 3, 5, 4, 7, 2, 2, 6, 6, 2, 2, 6, 6, 8, 9, 4, 3, 5, 4, 6, 1, 0, 4, 2, 3, 1, 0, 1, 5, 6, 7, 4, 3, 0, 0, 0, 7, 2, 8, 9, 4, 4, 7, 5, 7, 0, 8, 8, 2, 4, 7, 8, 0, 5, 5, 6, 9, 9, 5

Offset: 1

Jean-François Alcover, May 27 2014

			7.3719688014613165091531912...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.7 Lengyel's constant, p. 318.

G. C. Greubel, Table of n, a(n) for n = 1..2500

Cf. A048651, A086053, A242939, A182176.

digits = 100; EllipticTheta[3, 0, 1/2]/NProduct[1-2^(-j), {j, 1, Infinity}, WorkingPrecision -> digits + 10, NProductFactors -> digits] // RealDigits[#, 10, digits]& // First
RealDigits[EllipticTheta[3, 0, 1/2]/QPochhammer[1/2, 1/2], 10, 100][[1]] (* Vladimir Reshetnikov, Oct 17 2016 *)

th3(x)=1 + 2*suminf(n=1,x^n^2)
th3(1/2)/prodinf(n=1,1-2.^-n) \\ Charles R Greathouse IV, Jun 06 2016

(Sum_(k=-infinity..infinity) q^(-k^2)) / (prod_(j>0) (1-q^(-j))), with q = 2.

cp's OEIS Frontend

A182176 Number of affine subspaces of GF(2)^n.

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